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Quantum oscillations in the microwave magnetoabsorption of a
two-dimensional electron gas
Fedorych, O.M.; Potemski, M.; Studenikin, S.A.; Gupta, J.A.; Wasilewski,
Z.R.; Dmitriev, I.A.
Quantum oscillations in the microwave magnetoabsorption of a two-dimensional electron gas
O. M. Fedorych,1M. Potemski,1S. A. Studenikin,2J. A. Gupta,2Z. R. Wasilewski,2and I. A. Dmitriev3,
*
1Grenoble High Magnetic Field Laboratory, CNRS, Grenoble, France 2Institute for Microstructural Sciences, NRC, Ottawa, Ontario, Canada K1A-0R6 3Institute of Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
共Received 22 February 2010; published 4 May 2010兲
We report on the experimental observation of the quantum oscillations in microwave magnetoabsorption of a high-mobility two-dimensional electron gas induced by Landau quantization. Using original resonance-cavity technique, we observe two kinds of oscillations in the magnetoabsorption originating from inter-Landau-level and intra-Landau-level transitions. The experimental observations are in full accordance with theoretical pre-dictions. Presented theory also explains why similar quantum oscillations are not observed in transmission and reflection experiments on high-mobility structures despite of very strong effect of microwaves on the dc resistance in the same samples.
DOI:10.1103/PhysRevB.81.201302 PACS number共s兲: 73.50.Mx, 73.50.Jt, 78.20.Ls, 78.67.De
Quantum oscillations in absorption 共QMA兲 by a two-dimensional electron gas共2DEG兲 in perpendicular magnetic field B, governed by the ratio /c of the wave frequency
= 2fmw of external electromagnetic wave and the
cyclo-tron frequency c= eB / mc, were predicted long ago by
Ando1and observed experimentally2in the IR absorption on
a low-mobility and high-density 2DEG in a Si-inversion layer. Recently, similar/coscillations were discovered in the dc resistance of a high-mobility 2DEG irradiated by microwaves.3 Particularly intriguing are zero-resistance
states4 which develop in the minima of these
microwave-induced resistance oscillations共MIRO兲.
Theoretically, both MIRO共Refs.5–7兲 and QMA 共Refs.1
and5–8兲 stem from microwave-assisted transitions between disorder-broadened Landau levels共LLs兲. However, in experi-ments on high-mobility samples no QMA were observed so far despite strong MIRO showing up in the same experimen-tal conditions. Several attempts to measure microwave re-flection or transmission simultaneously with MIRO reported either single cyclotron resonance 共CR兲 peak9–11 or more
complex structure dominated by confined magnetoplasmons 共CMPs兲.12–14
Using original resonance-cavity technique, in this work we observe well-pronounced QMA in a high-mobility GaAs/ AlGaAs sample which also reveals strong MIRO in dc trans-port measurements. For both the T-independent QMA and dynamic Shubnikov-de Haas oscillations共SdHO兲, the experi-mental results fully agree with the theoretical predictions of Ref. 5, which generalizes theory1 of Ando for the case of
smooth disorder potential appropriate for high-mobility structures. In addition, we explain the failure to observe such quantum oscillations in transmission and reflection experi-ments.
We start with a summary of relevant theoretical results which includes QMA theory5 for dynamic conductivity at
high LLs and nonlinear relation between the absorption and dynamic conductivities8,10,15 specific for high-mobility
2DEG samples. Consider a plane wave normally incident to the 2DEG at the interface z = 0 between two dielectrics with permittivity⑀1共z⬍0兲 and⑀2共z⬎0兲. The electric field Re El
of external 共l=e兲, reflected 共l=r兲, and transmitted 共l=t兲
waves is a real part of El= Elexp共iklz− it兲兺⫾s⫾共l兲e⫾, where
the wave numbers ke/
冑
⑀1= −kr/冑
⑀1= kt/冑
⑀2=/c,coeffi-cients s⫾共l兲 describe the polarization, 兺⫾兩s⫾共l兲兩 2
= 1, and
冑
2e⫾= ex⫾ iey. According to the Maxwell equations,boundary conditions at z= 0 read Et= Er+ Ee and
z共Et− Er− Ee兲=共4/c2兲tˆ Et. It follows that:
冑
⑀1Ees⫾共e兲/Ets共t兲⫾=冑
⑀eff+ 2⫾/c, 共1兲where 2
冑
⑀eff=冑
⑀1+冑
⑀2. Further, ⫾=xx⫾ iyx are theei-genvalues of the complex conductivity tensor ˆ having the symmetriesxx=yy andxy= −yx, namely,ˆ e⫾=⫾e⫾.
Equation共1兲 yields the absorption A, transmission T, and reflection R coefficients 共see also Refs.8,10, and 15兲,
A =
兺
⫾冑
⑀1兩s⫾共e兲兩 2 兩冑
⑀eff+ 2⫾/c兩2 Re4⫾ c , 共2兲 T =兺
⫾冑
⑀1⑀2兩s⫾共e兲兩2 兩冑
⑀eff+ 2⫾/c兩2 , 共3兲 R =兺
⫾ 兩s⫾共e兲兩2冏
冑
⑀1−冑
⑀2− 4⫾/c冑
⑀1+冑
⑀2+ 4⫾/c冏
2 . 共4兲It is important to mention that the dynamic conductivity⫾, which is the focus of present study, is the response to the screened electric field acting on 2D electrons. By contrast, coefficients in Eqs. 共2兲–共4兲 represent a response to the 共un-screened兲 electric component of incoming wave and, there-fore, measure both single-particle 共transport兲 and collective 共screening兲 properties of 2DEG.
The dynamical screening, represented by the denomina-tors in Eqs. 共2兲–共4兲, becomes particularly strong in high-mobility structures where the ratio兩2⫾/c兩 reaches values
much larger than unity. Indeed, in the absence of Landau quantization the conductivity⫾=⫾
D
is given by the Drude formula
⫾D = ne 2/m tr −1 − i共⫾c兲 , 共5兲
wheretr is the momentum relaxation time. The absorption,
Eq. 共2兲, takes the form
AD=
冑
⑀1 ⑀eff兺
⫾ 兩s⫾共e兲兩 2 ⍀tr −1 共⍀ +tr−1兲2 +共⫾c兲2 , 共6兲where ប⍀ = 2␣F/
冑
⑀eff, ␣= e2/បc⯝1/137 is thefine-structure constant, and F is the Fermi energy of 2DEG. In
high-mobility 2DEG ⍀trⰇ 1 and the width of the cyclotron
peak in Eqs.共2兲–共4兲 and 共6兲 is dominated by strong reflection of microwaves. In the region兩−c兩ⱗ⍀, where 兩2−兩Ⰷc,
the collective effects are pronounced. In this region, a special care should be taken to avoid the finite-size magnetoplasmon effects.12–14
According to Ref.5共which generalizes the results of Ref.
1for the relevant case of smooth disorder potential, see also Refs. 7 and 8兲, Landau quantization at high LLs modifies Drude formula to the form which we call the quantum Drude formula 共QDF兲 in what follows:
Re⫾= ne2 m
冕
d共f− f+兲˜共兲tr,B −1 共 + 兲 关tr,B−2共兲 + tr,B −2 共 +兲兴/2 + 共 ⫾c兲2 , 共7兲 where f is the Fermi distribution function. The Landauquantization leads to the oscillatory density of states共DOS兲,
共兲=共+c兲⬅0˜共兲, and to renormalization of the
trans-port relaxation time,tr,B共兲⬅tr/˜共兲, where0= m / 2ប2is
the zero-B DOS per spin orientation. In the limit of strongly overlapping LLs, cqⰆ 1, where q is the quantum
relax-ation time, the DOS is weakly modulated by magnetic field
˜共兲 = 1 − 2␦cos2
c
, ␦= e−/cqⰆ 1. 共8兲
In the opposite limit cqⰇ 1, LLs become separated,
˜共兲=qRe
冑
⌫2−共−n兲2, where ⌫ =冑
2c/q⬍c/2, andnmarks the position of the nearest LL.
At the CR=c, QDF关Eq. 共7兲兴 reads
−兩c==−
D
兩c=
冕
d⌰关共兲兴f− f+
, 共9兲
where the integration is over the regions with共兲⬎0. In the case of overlapping LLs, this produces the classical Drude result−=−
D
meaning that quantum effects in the vicinity of the resonance are absent. In the case of separated LLs,
cⰇ ⌫, the conductivity is reduced,−兩c==共2⌫/c兲−
D
. At the same time, the CR width increases from tr−1 to
tr−1
c/⌫.5,16
In what follows we consider the region 兩−c兩Ⰷtr −1
where one can safely neglect tr−1 in the denominator of Eq. 共6兲 and QDF, Eq. 共7兲, which gives
A/AD=
冕
df− f+ ˜共兲˜共 +兲. 共10兲
Equation 共10兲 is the key theoretical result for our study. It expresses the imbalance between the rates of absorption and emission of the microwave quanta, both proportional to the product of initial and final densities of states. The screening properties of a 2DEG at 兩⫾c兩Ⰷtr
−1
are solely determined by the nondissipative part of conductivity, Im⫾⯝ne2/m共⫾c兲ⰇRe⫾, which remains finite at tr−1→ 0. That is why the transmission and reflection coeffi-cients, Eqs. 共3兲 and 共4兲, in high-mobility structures are al-most independent of disorder scattering making it very diffi-cult to observe QMA in direct transmission and reflection experiments.9–13
Magnetoabsorption at two temperatures T = 2 and 0.5 K as given by Eqs. 共6兲 and 共10兲 is illustrated in Fig. 1 for high-mobility 2DEG similar to best structures used for MIRO measurements. The broadening of the envelope Drude peak 关Eq. 共6兲兴 is dominated by strong reflection of microwaves near the CR since in our example ⍀⬃Ⰷtr−1
. The Drude peak in Fig. 1 is modulated by the dynamic SdHO 共at cⲏ and T= 0.5 K兲 and by the
temperature-independent QMA with maxima at the CR harmonics
= Nc, which we discuss below.
At low temperature, XT= 22T /បcⱗ 1, the absorption
关Eq. 共10兲兴 manifests the dynamic SdHO, which in the case of overlapping LLs关Eq. 共8兲兴 are given by
A/AD = 1 − 4XT␦ sinh XT c 2sin 2 c cos2 c . 共11兲 The dynamic SdHO are exponentially suppressed at XTⰇ 1
similar to SdHO in the dc resistance. In the inset it is clearly seen that the dynamic SdHO are periodically modulated ac-cording to sin共2/c兲 with nodes at the CR harmonics.
At e−XTⰆ␦, i.e., T Ⰷ T
D= ប / 2q, SdHO 关Eq. 共11兲兴
be-come exponentially smaller than T-independent/c
oscil-lations of second order O共␦2兲 which represent QMA,
FIG. 1. 共Color online兲 Magnetoabsorption calculated using Eqs. 共6兲 and 共10兲 for two temperatures T=2 and 0.5 K. Frequency fmw= 60 GHz, density n= 1.8⫻ 1011 cm−2, mobility = 107 cm2/V s, and
q= 15 ps. The difference between traces at T= 0.5 and 2 K in the inset shows the T-dependent dynamic SdHO.
FEDORYCH et al. PHYSICAL REVIEW B 81, 201302共R兲 共2010兲
A/AD⯝ 具˜共兲˜共 +兲典 = 2␦2
cos2
c
. 共12兲 Here the angular brackets denote —averaging over the pe-riod c. Maxima of QMA seen in Fig.1 appear at integer
harmonics of the CR/c= N. The amplitude of QMA关Eq. 共12兲兴 becomes of order unity when the DOS modulation is pronounced, i.e., ␦⬃1.17
Since QMA lie in the microscopic origin of MIRO,5it is
instructive to compare the expression 共12兲 to results for MIRO in the same regime. For inelastic mechanism of MIRO,5,6the photoresistivityph in terms of Drude
resistiv-ityD reads ph D= 1 + 2␦ 2−4inP D 20 ␦22 c sin2 c , 共13兲 where the microwave power dissipated in the absence of Landau quantization is PD= AD
冑
⑀1cEe2/
4 andinis the
in-elastic relaxation time. Below we use this equation to deter-mineqfrom dc measurement of MIRO. The known value of q enables a direct comparison of the measured and
calcu-lated QMA关Eq. 共12兲兴 without fitting parameters.
Experiment.Two rectangular samples were cleaved from a single molecular beam epitaxy-grown wafer of a high-mobility GaAs/AlGaAs heterostructure V0050. After illumination with red light the electron concentration was n = 3.6⫻ 1011 cm−2 and mobility = 5 ⫻ 106 cm2/V s.
Sample 1 was prepared for dc transport measurements. In-Sn Ohmic contacts were made by rapid annealing in reducing atmosphere of argon bubbled through hydrochloric acid. The sample was placed in a helium cryostat equipped with a su-perconducting magnet. Similar to Refs. 10 and 12, micro-waves from an HP source, model E8257D, were delivered to the sample using Cu-Be coaxial cable terminated with a 3 mm antenna.
Inset in Fig. 2 shows dc resistance Rxx measured in
sample 1 at fmw= 70 GHz and T = 2 K which displays
well-resolved MIRO. Main panel in Fig. 2 presents the depen-dence of log10共⌬Rxxc/兲 vs /c, where ⌬Rxxis the
am-plitude of MIRO measured between adjacent peaks and dips.
In accordance with Eq. 共13兲 containing ␦2= exp共−2/
cq兲,
this dependence is linear. The slope gives the quantum relax-ation time q= 9.1 ps which we use for calculation of QMA
in sample 2.
Sample 2 was cleaved from the same wafer in vicinity to sample 1. In order to reduce the CMP effects,14,15,18 which can obscure weak QMA oscillations under the investigation, we cleaved a narrow 0.5⫻ 1.5 mm2 rectangular stripe. The
magnetoabsorption experiment was performed using a home-built microwave cavity setup at liquid-helium temperatures.19 The cavity with a tunable resonance fre-quency had a cylindrical shape with 8 mm diameter and the height between 3 and 8 mm adjustable with a movable plunger. The cavity operated in TE011mode, where兵011其 are
the numbers of half-cycle variations in the angular, radial, and longitudinal directions, respectively. The 2DEG stripe was placed at the bottom of the cavity with the external magnetic field normal to the 2DEG plane and the microwave electric field of the TE011mode oriented along the short side
of the rectangular sample. The sample was placed in a “face up” fashion such that the active 2DEG layer is separated from the plunger surface by the substrate. This geometry has higher sensitivity to weak-absorption signals such as QMA but is not appropriate to study, i.e., CMPs in the vicinity of the CR due to cavity over-coupling effects. To further im-prove sensitivity we measure the differential signal with re-spect to magnetic field.
The top curve in Fig. 3 presents B dependence of the second derivative of the measured absorption for fmw= 58.44 GHz whereas the bottom curve shows the
sec-ond B derivative of the absorption coefficient A, Eqs.共6兲 and 共10兲, calculated without fitting parameters using q= 9.1 ps
determined from MIRO measurements on sample 1共Fig.2兲. Both curves demonstrate well-resolved QMA with maxima at harmonics of the CR and dynamic SdHO with maxima determined by the position of the chemical potential with respect to LLs. Clearly, the theory reproduces the experimen-tal trace in Fig. 3 quite well everywhere except for the shaded region around the CR where the signal is distorted due to the CMP absorption. The dimensions of sample 2 FIG. 2. 共Color online兲 MIRO measured on sample 1 for
fmw= 67 GHz at T = 2 K 共inset兲, and the amplitude of MIRO vs / c in semilog scale 共main panel兲. The linear fit yields the
quantum-scattering time q= 9.1 ps, see Eqs.共8兲 and 共13兲.
FIG. 3. 共Color online兲 Magnetoabsorption measured on sample 2 at T = 2 K for fmw= 58.44 GHz共top curve兲 and absorption coef-ficient A关Eqs. 共6兲 and 共10兲, bottom curve兴 calculated without fitting parameters using q= 9.1 ps determined from MIRO measurements on sample 1共Fig.2兲.
were chosen to allow only one CMP mode at B = 0.092 T. The absence of higher modes for fmw⬍ 63 GHz enabled the
possibility to observe clear quantum oscillations on both sides of the shaded region where finite-size effects14,15,18are
not essential.
The low-field traces共c/⬍ 1 / 2兲 of the
magnetoabsorp-tion are shown in Fig. 4 for several microwave frequencies together with the function␦2cos共2/
c兲. The phase and B
damping of the observed QMA follow well the theoretical
dependence, Eq.共12兲, without fitting parameters. We, there-fore, believe that the observed oscillations are indeed QMA predicted in Refs. 1and5, which provides an important ex-perimental evidence supporting the theory of MIRO 共Refs. 5–7兲 based on inter-LL transitions. In our sample, QMA are strongly damped 关␦2
= 0.02 at c=/2,
/2= fmw= 58.44 GHz, and q= 9.1 ps, see Eq. 共12兲兴 which makes their observation difficult. We expect that much stronger QMA as well as dynamic SdHO can be observed on samples with higher mobility 共longer q兲, as simulated
in Fig. 1, provided the CMP effects are avoided or sufficiently reduced.
In summary, we have observed quantum magneto-oscillations in the microwave absorption and dynamic SdHO in a high-mobility 2DEG. For this purpose we used a sensi-tive high-Q cavity technique and developed a special setup to avoid undesirable magnetoplasmon effects masking the quantum oscillations. Using the quantum Drude formula5
and the quantum relaxation time extracted from the MIRO measurements on the same wafer we were able to reproduce the experimental results for absorption without fitting param-eters, which provides a strong experimental support to the theory of MIRO and QMA based on inter-LL transitions.
We are thankful to A. D. Mirlin, D. G. Polyakov, B. I. Shklovskii, S. A. Vitkalov, and M. A. Zudov for fruitful dis-cussions. This work was supported by the DFG, by the DFG-CFN, by Rosnauka under Grant No. 02.740.11.5072, by the RFBR, and by the NRC-CNRS project.
*Also at Ioffe Physical Technical Institute, 194021 St. Petersburg, Russia.
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17Resonant features at overtones of CR can be induced by the quasiclassical memory effects even in the absence of Landau quantization,␦→ 0关see I. A. Dmitriev, A. D. Mirlin, and D. G. Polyakov, Phys. Rev. B 70, 165305 共2004兲兴. These classical oscillations originate from the non-Markovian memory effects in the presence of both long-range and short-range randomness in the static impurity potential. The predicted phase of classical oscillations in the dynamic conductivity 共and therefore in the absorption coefficient兲 is opposite to the phase of quantum os-cillations 关Eq. 共12兲兴 while the amplitude can be comparable to the value of Drude conductivity.
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compared to the theory关Eq. 共12兲, parameters as in Fig.3兴.
FEDORYCH et al. PHYSICAL REVIEW B 81, 201302共R兲 共2010兲
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